$A$ ladder $5 \ m$ long rests against a vertical wall. If its top slides downwards at the rate of $10 \ cm/s$,then the angle between the ladder and the floor is decreasing at the rate of how many $rad/s$ when its lower end is $4 \ m$ away from the wall?

The distance $s$ in meters travelled by a particle in $t$ seconds is given by $s = \frac{2 t^3}{3} - 18 t + \frac{5}{3}$. The acceleration when the particle comes to rest is (in $m/s^2$)

$A$ water tank has the shape of an inverted right circular cone whose semi-vertical angle is $\tan^{-1}\left(\frac{1}{2}\right)$. Water is poured into it at a constant rate of $5 \text{ m}^3/\text{min}$. The rate at which the level of water is rising in $\text{m/min}$ at the instant when the depth of water in the tank is $10 \text{ m}$ is:

If the volume of a sphere increases at the rate of $2 \pi \text{ cm}^3/\text{s}$,then the rate of increase of its radius (in $\text{cm}/\text{s}$),when the volume is $288 \pi \text{ cm}^3$,is

$A$ tank in the shape of a rectangular parallelepiped has a volume of $27 \ m^3$. This tank is filled with water such that the rate of change of the water level is thrice the rate of change of the water quantity falling into the tank. The height of the tank (in meters) is:

If the velocity $v$ of a particle moving on a straight line is proportional to the cube root of its displacement $x$,then its acceleration $a$ is